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Math  /  Algebra

QuestionGiven the function C(r)=2r4−10r3−12r2C(r)=2 r^{4}-10 r^{3}-12 r^{2} its CC-intercept is □\square its rr-intercepts are □\square

Studdy Solution

STEP 1

1. We are given the function C(r)=2r4−10r3−12r2 C(r) = 2r^4 - 10r^3 - 12r^2 .
2. We need to find the C C -intercept, which is the value of C(r) C(r) when r=0 r = 0 .
3. We need to find the r r -intercepts, which are the values of r r when C(r)=0 C(r) = 0 .

STEP 2

1. Find the C C -intercept by evaluating C(r) C(r) at r=0 r = 0 .
2. Find the r r -intercepts by solving the equation C(r)=0 C(r) = 0 .

STEP 3

To find the C C -intercept, substitute r=0 r = 0 into the function:
C(0)=2(0)4−10(0)3−12(0)2 C(0) = 2(0)^4 - 10(0)^3 - 12(0)^2

STEP 4

Simplify the expression:
C(0)=0−0−0 C(0) = 0 - 0 - 0 C(0)=0 C(0) = 0
The C C -intercept is 0 \boxed{0} .

STEP 5

To find the r r -intercepts, solve the equation:
2r4−10r3−12r2=0 2r^4 - 10r^3 - 12r^2 = 0
Factor out the greatest common factor, which is 2r2 2r^2 :
2r2(r2−5r−6)=0 2r^2(r^2 - 5r - 6) = 0

STEP 6

Set each factor equal to zero:
1. 2r2=0 2r^2 = 0
2. r2−5r−6=0 r^2 - 5r - 6 = 0
Solve the first equation:
2r2=0 2r^2 = 0 r2=0 r^2 = 0 r=0 r = 0

STEP 7

Solve the second equation using the quadratic formula:
The quadratic formula is r=−b±b2−4ac2a r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} .
For r2−5r−6=0 r^2 - 5r - 6 = 0 , a=1 a = 1 , b=−5 b = -5 , c=−6 c = -6 .
r=−(−5)±(−5)2−4(1)(−6)2(1) r = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-6)}}{2(1)} r=5±25+242 r = \frac{5 \pm \sqrt{25 + 24}}{2} r=5±492 r = \frac{5 \pm \sqrt{49}}{2} r=5±72 r = \frac{5 \pm 7}{2}
This gives two solutions:
r=122=6 r = \frac{12}{2} = 6 r=−22=−1 r = \frac{-2}{2} = -1
The r r -intercepts are 0,−1,6 \boxed{0, -1, 6} .

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